​ Show that the magnetic field at any point on the axis of the solenoid having n turns per unit length is B =1/2 µ0 nI (cos θ1 – cosθ2 ) ​

Question

Show that the magnetic field at any point on the axis of the solenoid having n turns per unit length is B = \(\frac{1}{2}\)µ₀ nI (cos θ₁ – cosθ₂ )

Answer 1

A solenoid is a cylindrical coil having number of circular turns. Consider a solenoid having radius R consists of n number’of turns per unit length. Let ‘P’ be the point at a distance ‘x’ from the origin of the solenoid. The current carrying element dx at a distance x from origin and the distance r from point ‘P’

\(\sqrt{R^2 + (x_0 -x)^2}\)

The magnetic field due to current carrying circular coil at any axis is

Now total magnetic field can be obtained by integrating from θ₁ to θ₂ , we get